An entity may seek to monitor changes in an attribute associated with members of a population. The members may be individuals or things. The attribute may be quantified for each member of the population. For example, each member may be “scored” with respect to the attribute. The score may indicate the extent to which, or the likelihood that, the member possesses the attribute. The members' scores may change over time.
One way to characterize the presence of the attribute at a selected time is to formulate a distribution showing how likely the attribute is to occur in portions of the population. The entity may desire to modify its own behavior based on observed changes in the distribution. The entity may use an objective index, such as a population stability index, to quantify the change.
For example, the entity may seek to monitor an attribute such as an economic behavior of the entity's customers. In the financial services industry, for example, financial services companies often monitor credit-related behavior of a population of its customers.
One form of credit-related behavior may include when a customer accumulates debt on a credit account and cannot repay the debt. A financial services company may seek to monitor risk that is associated with a population of its customers. The population may include all or a portion of its customers. A change in conditions, such as economic conditions, political conditions, weather conditions and the like, may modify the risk associated with the population.
Table 1 shows a typical use of a population stability index to characterize a change between a typical model score distribution based on a typical development data set and the model score distribution based on a typical validation data set. The model score indicates likelihood of inability to pay off a debt based on a credit behavior model.
TABLE 1Population Stability Index (“PSI”) .Model scoreModel scoredistribution for adistribution for aScore rangeDeciledevelopment datasetvalidation datasetcontribution110%20% 0.069314718210%9%0.001053605310%9%0.001053605410%9%0.001053605510%9%0.001053605610%9%0.001053605710%9%0.001053605810%9%0.001053605910%9%0.00105360510 10%8%0.004462871Total100% 100% 0.08220643 (=PSI1)1Population Stability Index may, in some disciplines, be referred to as “Information Value.” 
The deciles in Table 1 define a distribution of a model score among the members of a customer population. The model score corresponds to the likelihood of inability to pay a debt. Decile 1 represents the highest likelihood (Model Score=91-100%) of inability to pay a debt. Decile 10 represents the lowest likelihood of inability to pay a debt (Model Score=1-10%).
The model score distribution in a development data set shows, for each decile, the percentage of the population at a first time that received a score corresponding to the decile.
The model score distribution for a validation data set shows, for each decile, the percentage of the population at a second time that received a score corresponding to the decile. The most noticeable change is a doubling of the portion of the population that scored in the highest decile.
Equation 1 shows how score range contribution is evaluated, for each decile i, from the development distribution and the validation distribution.
                              Score          ⁢                                          ⁢          Range          ⁢                                          ⁢                      Contribution            i                          =                              (                                          Validation                ⁢                                                                  ⁢                                  %                  i                                            -                              Development                ⁢                                                                  ⁢                                  %                  i                                                      )                    ·                      Ln            ⁡                          (                                                Validation                  ⁢                                                                          ⁢                                      %                    i                                                                    Development                  ⁢                                                                          ⁢                                      %                    i                                                              )                                                          Eqn        .                                  ⁢        1            
Table 1 shows the score range contribution for each of the i deciles. (A decile includes 10 percentage points that define a score range.) Validation % for each decile is taken from the model score distribution from the validation data set. Development % for each decile is taken from the model score distribution from the development data set.
Equation 2 shows how Population Stability Index is evaluated from the score range contributions of the 10 deciles.
                              P          ⁢                                          ⁢          S          ⁢                                          ⁢          I                =                              ∑                          i              =              1                        10                    ⁢                      Score            ⁢                                                  ⁢            Range            ⁢                                                  ⁢                          Contribution              i                                                          Eqn        .                                  ⁢        2            
PSI for the example in Table 1, as shown at the bottom of Table 1, is 0.08220643.
A greater PSI value indicates a larger shift in a distribution (e.g., between the development and validation distributions). Typically, in the financial services industry, widely accepted threshold PSI values are believed to correspond to different levels of population stability. Table 2 shows those PSIs and corresponding stability levels.
TABLE 2PSIs and corresponding stability levels.PSIStability level<0.1Stable0.1-0.3Slight shift>0.3Significant shift
The PSI in the example shown in Table 1 is less than 0.1. According to the scheme shown in Table 2, such a PSI indicates that the population characterized in Table 1 is stable. If the entity's decisions are based on the Table 2 scheme, then, the large increase (a doubling) of the decile 1 proportion in the validation distribution (relative to the development distribution) would go unnoticed. This would be particularly true if the Table 2 scheme were the basis for an automatic stability alert that is generated by a computational machine. The Table 2 scheme therefore may be inadequate for the identification of population shifts that are economically meaningful.
Also, the Table 2 scheme is typically used in the context of six- to 12-month population shifts. Shifts over such a time scale may be large enough to generate PSIs that are sufficient to cause the Table 2 scheme to indicate slight to significant shifts. When analyzing shifts over a shorter time period, however, even economically significant shifts may not generate PSIs that are large enough to cause the Table 2 scheme to indicate slight to significant shifts and may therefore be overlooked.
It would be desirable, therefore, to provide apparatus and methods for selecting PSI values that correspond to population stability.